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Abstrakty
We present classical and generalized Riemann-Hilbert problem in several contexts arising from K-theory and bordism theory. The language of Fredholm pairs turns out to be useful and unavoidable. We propose an abstract formulation of a notion of bordism in the context of Hilbert spaces equipped with splittings.
Wydawca
Rocznik
Tom
Strony
479--496
Opis fizyczny
Bibliogr. 30 poz., rys.
Twórcy
autor
- Institute of Mathematics Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland
autor
- Institute of Mathematics, Warsaw University, Banacha 2, 02-097, Warszawa, Poland
Bibliografia
- [1] M. F. Atiyah, K-theory. Lecture notes by D. W. Anderson, Benjamin, New York Amsterdam 1967.
- [2] M. F. Atiyah, I. M. Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc., 69 (1963) 422-433.
- [3] M. F. Atiyah, V. K. Patodi, I. M. Singer, Spectral asymmetry and Riemannian geometry. I., Math. Proc. Cambridge Philos. Soc., 77 (1975) 43-69.
- [4] B. Bojarski. The abstract linear conjugation problem and Fredholm pairs of subspaces, in: Differential and integral equations. Boundary value problems, Publications of I. N. Vekua Institute of Applied Mathematics, Tbilisi (1979) 45-60.
- [5] B. Bojarski, Connections between complex and global analysis: some analytical and geometrical aspects of the Riemann-Hilbert transmission problem, Complex Analysis, Math. Lehrbücher Monogr. II. Abt. Math. Monogr. 61, Akademie-Verlag, Berlin (1983) 97-110.
- [6] B. Bojarski, The geometry of the Riemann-Hilbert problem in: Geometric aspects of partial differential equations. Proceedings of a minisymposium on spectral invariants, heat equation approach. Roskilde, Denmark, September 18-19, 1998, eds.: Booss-Bavnbek, Bernhelm et al., Contemp. Math., 242 (1999) 25-33.
- [7] B. Bojarski, The geometry of the Riemann-Hilbert problem II, Boundary value problems, integral equations and related problems (Beijing/Chengde, 1999) World Sci. Publishing. River Edge, NJ (2000) 41-48.
- [8] B. Bojarski, A. Weber, Riemann-Hilbert problem: K-theory and bordisms, in preparation.
- [9] B. Booss-Bavnbek, K. P. Wojciechowski, Desuspension of splitting elliptic symbols I, Ann. Global Anal. Geom., 3 (3) (1985) 337-383; II, Ann. Global Anal. Geom., 4 (3) (1986) 349-400.
- [10] B. Booss-Bavnbek, K. P. Wojciechowski, Elliptic boundary problems for Dirac operators, Birkhäuser, Boston 1993.
- [11] L. Boutet de Monvel, Problème de Riemann-Hilbert, Mathématique et Physique, (Séminaire de l’ENS 79-82). Progr. in Math., 37 (1983) 281-288, 299-306.
- [12] A. Connes, Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math., 62 (1985) 257-360.
- [13] A. Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA 1994.
- [14] X. Dai, W. Zhang, Splitting of the family index, Comm. Math. Phys., 182 (2) (1996) 303-317.
- [15] F. D. Gakhov, Boundary value problems, Nauka, Moscow 1977.
- [16] I. T. Gohberg, M. G. Krein, Introduction to the theory of linear nonselfadjoint operators in Hilbert space, Nauka, Moscow 1965.
- [17] I. T. Gohberg, N. Y. Krupnik, Introduction to the theory of one-dimensional singular integral operators (in German), Birkhäuser, Basel Boston 1979.
- [18] G. G. Kasparov, Topological invariants of elliptic operators. I. K-homology., Math. USSR-Izv., 9 (4) (1975) 751-792; translated from Izv. Akad. Nauk SSSR Ser. Mat., 39 (4) (1975) 796-838.
- [19] T. Kato, Perturbation theory for linear operators, Grundlehren Math. Wiss., 132, Springer-Verlag, New York 1966.
- [20] N. H. Kuiper, The homotopy type of the unitary group of Hilbert space, Topology, 3 (1965) 19-30.
- [21] G. S. Litvinchuk, Boundary value problems and singular integral equations with a shift, Nauka, Moscow 1977.
- [22] G. S. Litvinchuk, I. M. Spitkovski, Factorisation of measurable matrix functions, Birkhäuser, Basel 1987.
- [23] S. G. Mikhlin, S. Prössdorf, Singular integral operators (in German), Mathematische Monographien, 52, Akademie-Verlag, Berlin 1980.
- [24] J. Milnor, Lectures on the h-cobordism theorem, Princeton University Press, Princeton, N. J. 1965.
- [25] N. I. Muskhelisvili, Singular integral equations, Nauka, Moscow 1968.
- [26] A. Pressley, G. B. Segal, Loop groups, Oxford Math. Monogr., Oxford Univ. Press, New York 1986.
- [27] S. G. Scott, K. P. Wojciechowski, The ζ-determinant and Quillen determinant for a Dirac operator on a manifold with boundary, Geom. Funct. Anal., 10 (2000) 1202-1236.
- [28] G. B. Segal, Topological field theory (Stanford notes), available at the web page http://www.cgtp.duke.edu/ITP99/segal/.
- [29] V. I. Shevchenko, On the Hilbert problem for holomorphic vector functions in space (in Russian), in: Differential and integral equations. Boundary value problems, Publications of I. N. Vekua Institute of Applied Mathematics, Tbilisi (1979) 279-299.
- [30] L. F. Zverovich, Riemann problem for vector functions for general contours on a Riemann surface, Siberian Math. J., 19 (3) (1975) 510-519.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-BAT2-0001-0575